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Properties

Label 2-825-165.38-c1-0-28
Degree $2$
Conductor $825$
Sign $-0.0844 - 0.996i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0763 + 0.149i)2-s + (1.62 + 0.602i)3-s + (1.15 + 1.59i)4-s + (−0.214 + 0.197i)6-s + (−0.585 + 3.69i)7-s + (−0.659 + 0.104i)8-s + (2.27 + 1.95i)9-s + (−0.807 − 3.21i)11-s + (0.921 + 3.28i)12-s + (−0.0739 + 0.145i)13-s + (−0.508 − 0.369i)14-s + (−1.18 + 3.64i)16-s + (4.68 − 2.38i)17-s + (−0.466 + 0.191i)18-s + (1.80 − 2.48i)19-s + ⋯
L(s)  = 1  + (−0.0539 + 0.105i)2-s + (0.937 + 0.347i)3-s + (0.579 + 0.797i)4-s + (−0.0873 + 0.0805i)6-s + (−0.221 + 1.39i)7-s + (−0.233 + 0.0369i)8-s + (0.758 + 0.651i)9-s + (−0.243 − 0.969i)11-s + (0.266 + 0.949i)12-s + (−0.0205 + 0.0402i)13-s + (−0.135 − 0.0987i)14-s + (−0.295 + 0.910i)16-s + (1.13 − 0.579i)17-s + (−0.109 + 0.0451i)18-s + (0.414 − 0.570i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0844 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0844 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.0844 - 0.996i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.0844 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54544 + 1.68197i\)
\(L(\frac12)\) \(\approx\) \(1.54544 + 1.68197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 - 0.602i)T \)
5 \( 1 \)
11 \( 1 + (0.807 + 3.21i)T \)
good2 \( 1 + (0.0763 - 0.149i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (0.585 - 3.69i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (0.0739 - 0.145i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-4.68 + 2.38i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-1.80 + 2.48i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (3.73 - 3.73i)T - 23iT^{2} \)
29 \( 1 + (3.63 - 2.64i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.60 + 4.94i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.30 - 8.21i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-6.69 + 9.21i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (5.81 + 5.81i)T + 43iT^{2} \)
47 \( 1 + (-1.01 - 6.40i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-3.24 - 1.65i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-7.96 + 5.78i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.302 + 0.932i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.19 + 2.19i)T - 67iT^{2} \)
71 \( 1 + (11.4 + 3.71i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.355 - 0.0562i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-8.37 + 2.72i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.41 + 10.6i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 7.69T + 89T^{2} \)
97 \( 1 + (-5.52 - 2.81i)T + (57.0 + 78.4i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.31990687853472632594381616611, −9.288552994329814076305621599241, −8.807521767529782739325187857499, −7.938144154677260886016038630177, −7.33936826443495275514497479960, −6.04230324182305228345679928696, −5.22020337127095109564509760589, −3.63605582816736933072204738767, −3.02199924859163362146997645687, −2.10889776381343168856752568104, 1.08233608593671432341211516990, 2.13157359274962143130432815262, 3.43513816440344255350593139531, 4.36253761564007869378256870176, 5.73623137637559229998528472267, 6.79839835027027051509685677798, 7.41106769850271446602850444000, 8.081861702811405434697397394158, 9.413680507601321624796077709893, 10.19655934976935524770142283688

Graph of the $Z$-function along the critical line